Velocity-independent $\tau$ -$p$ moveout in a horizontally-layered VTI medium

Field data example

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Figure 8. $\tau$ -$p$  or Radon transformed data from a marine acquisition (a). The data are fairly clean even though there is a slight decrease in SNR for later and steepest events. The CMP maximum offset-to-depth ratio reaches 1.5 for larger value of the horizontal slope $p$ . Dominant local-slope field (b) measured using PWD algorithm. Using the slopes we estimate the zero-slope time $\tau _0$ mapping fields (c) that predicts reflection curves by which we flatten the original gather (d).

Figure 9 presents the results of the proposed $\tau$ -$p$  processing on a field data example from a marine acquisition. Figure 9a shows a $\tau$ -$p$  transformed CMP gather. The data are taken from a deep water (3.5 s of sea-depth) dataset with poor offset sampling ( $\sim 100 m$ ) that aliases the steepest seismic events. Spatial aliasing creates artifacts in $\tau$ -$p$  domain that bias the PWD slope estimate. In order to mitigate the effect of the aliasing, we interpolated the raw data by means of an FX algorithm (Spitz, 1991). The original trace recording is 7.0 s long but, since the SNR decreases significantly after 5.0 s, we window the CMP gather and process seismic events only between 3.0 and 6.0 s. The CMP maximum offset-to-depth ratio reaches 1.5 for larger value of the horizontal slope $p$ . As for the synthetic case, the data should carry enough information to well resolve the horizontal velocity and the anellipticity parameter. Figure 9b shows the dominant local-slope $R$ field automatically measure from the data using the PWD algorithm. As in the synthetic case, we use these slopes to construct the prediction operator that allows us to paint the zero-slope traveltime map $\tau _0$ along the reflection events (Figure 9c). The $\tau _0$ values are finally used to unwrap the trace shifts until the gather is completely flattened. The good alignment of the NMO corrected traces (Figure 9d) confirms the robustness of predictive painting with real data. Figure 10 shows spectra of the recovered interval parameters using Fowler's equations 27 and 28. The plots are overlaid with the profiles (yellow curves) recovered using a layer-based $t$ -$X$  Dix inversion (Ferla and Cibin, 2009) and by the profiles obtained after an automated picking of the recovered trends. Our solution (red curves) follows the Dix trends (yellow curves) even though it exhibits a slight decrease in accuracy. The poor SNR for later and steeper events and the numerical differentiation of the zero-slope traveltime $\tau _0$ and slope $R$ make the field data results noisier. As expected, the high-order moveout parameters appear to be more sensitive to the noise. Moreover, the more pronounced enlargement of the $\hat{\eta}$ trend in comparison with the $\hat{V}_H$ trend confirms that the latter parameter is better constrained by the data (Tsvankin, 2006).

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Figure 9. Parameters spectra for interval normal moveout (a), horizontal velocity (b) and anellipticity parameter $\eta$ (c). These spectra result from the application of Fowler's equations using painted $\tau _0$ field. The red lines are the profiles after an automated picking of the estimated spectra. The yellow lines are the recovered profiles after a layer-based $t$ -$X$  Dix inversion procedure (Ferla and Cibin, 2009).

Velocity-independent $\tau$ -$p$ moveout in a horizontally-layered VTI medium